HW2S24_Solutions (1)
.pdf
keyboard_arrow_up
School
San Jose State University *
*We aren’t endorsed by this school
Course
102
Subject
Industrial Engineering
Date
Apr 3, 2024
Type
Pages
7
Uploaded by ColonelLightning4050
EE250: Probability, Random Variables and
Stochastic Processes
Spring 2024
Homework 2 Solutions
Due
11:59PM,
Feb. 27
th
2024
Notes:
a. Each submission must be a single pdf file.
b. Solutions should be submitted on Canvas.
c. This assignment is worth 100 points.
1.
Monitor three consecutive packets going through an Internet router. Classify each one
as either video (v) or data (d). Your observation is a sequence of three letters (each one is
either v or d). For example, three video packets corresponds to vvv. The outcomes vvv
and ddd each have probability 0.2 whereas each of the other outcomes vvd, vdv, vdd, dvv,
dvd, and ddv has probability 0.1. Count the number of video packets Nv in the three
packets you have observed. Calculate the following probabilities:
(a) P[Nv=2]
(b) P[Nv≥1]
(c) P[{vdv}|Nv=2]
(d) P[{ddd}|Nv=1]
(e) P[Nv=2| Nv≥1]
(f) P[Nv≥1| Nv=2]
2. A ternary communication system is shown in Figure below. Suppose that input
symbols 0, 1, and 2 occur with probability 1/3 respectively.
(a) Find the probabilities of the output symbols, i.e., P[the output symbol=i], i=0,1,2
(b) Suppose that a 1 is observed at the output.What is the probability that the input was
0? 1? 2?
(a)
P[the output symbol=0]=P[the output symbol=0| the input symbol=0]*P[the input
symbol=0]+ P[the output symbol=0| the input symbol=1]*P[the input symbol=1]+ P[the
output symbol=0| the input symbol=2]*P[the input symbol=2]=(1-ε)*1/3+ 0*1/3+ ε*1/3
=1/3
P[the output symbol=1]=P[the output symbol=1| the input symbol=0]*P[the input
symbol=0]+ P[the output symbol=1| the input symbol=1]*P[the input symbol=1]+ P[the
output symbol=1| the input symbol=2]*P[the input symbol=2]= ε*1/3+(1- ε)*1/3+ 0*1/3
=1/3
P[the output symbol=2]=P[the output symbol=2| the input symbol=0]*P[the input
symbol=0]+ P[the output symbol=2| the input symbol=1]*P[the input symbol=1]+ P[the
output symbol=2| the input symbol=2]*P[the input symbol=2]= 0+(1- ε)*1/3+ ε*1/3
=1/3
(b)
P[the input symbol=0| the output symbol=1]=
P[the output symbol=1| the input symbol=0]∗P[the input symbol=0]
±[the output symbol=1]
=
1
²
∗ε
1t²
= ε
P[the input symbol=1| the output symbol=1]=
P[the output symbol=1| the input symbol=1]∗P[the input symbol=1]
±[the output symbol=1]
=
1
²
∗³1− ε´
1t²
= 1 − ε
P[the input symbol=2| the output symbol=1]=
P[the output symbol=2| the input symbol=1]∗P[the input symbol=2]
±[the output symbol=1]
=
1
²
∗0
1t²
= 0
3. One of two coins is selected at random (equally likely to be selected) and tossed three
times. The first coin comes up heads with probability p1=2/3 and the second coin with
probability p2=1/3
(a) What is the probability that the number of heads is 3?
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help